What do the radiative decays of X(3872) tell us

Since the discovery of X(3872), its structure has been in ceaseless dispute. The data of $X(3872)\to {\pi }^{+}{\pi }^{-}{\pi }^{0}J\mathrm{/}\psi$ suggest that X(3872) may be a high-spin charmonium-like of $2^{-+}$. In terms of the light front quark model (LFQM) we calculate the rates of the radiative decays $X(3872)\to J\mathrm{/}\psi ({\psi }^{\mathrm{\prime }})\gamma$ supposing X(3872) to be a $2^{-+}$ charmonium. Within this framework, our theoretical prediction on $\mathcal{B}\mathcal{R}(X(3872)\to \psi (1S)\gamma )$ is at order of ${10}^{-3}$ which is slightly lower than the Babar’s data but close to the Belle’s. Our prediction on $\mathcal{B}\mathcal{R}(X(3872)\to {\psi }^{\mathrm{\prime }}\gamma )$ is at order of ${10}^{-5}$ if ${\psi }^{\mathrm{\prime }}$ is a pure 2S state or ${10}^{-4}$ if ${\psi }^{\mathrm{\prime }}$ is a $2S-1D$ mixture, which does not conflict with the upper bound set by the Belle collaboration, but is much lower than the Babar’s data. Thus if the future measurement decides the branching ratio of $\mathcal{B}\mathcal{R}(X(3872)\to {\psi }^{\mathrm{\prime }}\gamma )$ to be much larger than ${10}^{-4}$, the $2^{-+}$ assignment for X(3872) should be ruled out.